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I was reading Dieudonne's "On the history of the Weil conjectures" and found two things that surprised me. Dieudonne makes some assertions about the work of Artin and Schmidt which are no doubt correct, but he doesn't give references, and the thought of ploughing through Artin's collected works seems a bit daunting to me, so I thought I'd ask here first.

Background.

If $V$ is a smooth (affine or projective) curve over a finite field $k$ of size $q$, then $k$ has (up to isomorphism) a unique extension $k_n$ of degree $n$ over $k$ (so $k_n$ has size $q^n$) and one can define $N_n$ to be the size of $V(k_n)$. Completely concretely, one can perhaps imagine the case where $V$ is defined by one equation in affine or projective 2-space, for example $y^2=x^3+1$ (note that this equation will give a smooth curve in affine 2-space for $p$, the characteristic of $k$, sufficiently large), and simply count the number of solutions to this equation with $x,y\in k_n\ $to get $N_n$. Let $F_V(u)=\sum_{n\geq1}N_nu^n$ denote the formal power series associated to this counting function.

Now it turns out from the "formalism of zeta functions" that this isn't the most ideal way to package the information of the $N_n$, one really wants to be doing a product over closed points of your variety. If $C_d$ is the number of closed points of $V$ of degree $d$, that is, the number of closed points $v$ of (the topological space underlying the scheme) $V$ such that $k(v)$ is isomorphic to $k_d$, then one really wants to define $$Z_V(u)=\prod_{d\geq1}(1-u^d)^{-C_d}.$$ If one sets $u=q^{-s}$ then this is an analogue of the Riemann zeta function, which is a product over closed points of $Spec(\mathbf{Z})$ of an analogous thing.

Now the (easy to check) relation between the $C$s and the $N$s is that $N_n=\sum_{d|n}dC_d$, and this translates into a relation between $F_V$ and $Z_V$ of the form $$uZ_V'(u)/Z_V(u)=F_V(u).$$

This relation also means one can compute $Z$ given $F$: one divides $F$ by $u$, integrates formally, and then exponentiates formally; this works because $f'/f=(\log(f))'$.

The reason I'm saying all of this is just to stress that this part of the theory is completely elementary.

The Weil conjectures in this setting.

The Weil conjectures imply that for $V$ as above, the power series $Z_V(u)$ is actually a rational function of $u$, and make various concrete statements about its explicit form (and in particular the location of zeros and poles). Note that they are usually stated for smooth projective varieties, but in the affine curve case one can take the smooth projective model for $V$ and then just throw away the finitely many extra points showing up to see that $Z_V(u)$ is a rational function in this case too.

How to prove special cases in 1923?

OK so here's the question. It's 1923, we are considering completely explicit affine or projective curves over explicit finite fields, and we want to check that this power series $Z_V(u)$ is a rational function. Dieudonne states that Artin manages to do this for curves of the form $y^2=P(x)$ for "many polynomials $P$ of low degree". How might we do this? For $P$ of degree 1 or 2, the curve is birational to projective 1-space and the story is easy. For $V$ equals projective 1-space, we have $$F_V(u)=(1+q)u+(1+q^2)u^2+(1+q^3)u^3+\ldots=u/(1-u)+qu/(1-qu)$$ from which it follows easily from the above discussion that $$Z_V(u)=1/[(1-u)(1-qu)].$$ For polynomials $P$ of degree 3 or 4, the curve has genus 1 and again I can envisage how Artin could have approached the problem. The curve will be birational to an elliptic curve, and it will lift to a characteristic zero curve with complex multiplication. The traces of Frobenius will be controlled by the corresponding Hecke character, a fact which surely will not have escaped Artin, and I can believe that he was now smart enough to put everything together.

For polynomials of degree 5 or more, given that it's 1923, the problem looks formidable.

Q1) When Dieudonne says that Artin verified (some piece of) the Weil conjectures for "many polynomials of low degree", does he mean "of degree at most 4", or did Artin really move into genus 2?

How much further can we get in 1931?

Now this one really surprised me. Dieudonne claims that in 1931 F. K. Schmidt proved rationality of $Z_V(u)$, plus the functional equation, plus the fact that $Z_V(u)=P(u)/(1-u)(1-qu)$, for $V$ an arbitrary smooth projective curve, and that he showed $P(u)$ was a polynomial of degree $2g$, with $g$ the genus of $V$. This is already a huge chunk of the Weil conjectures. We're missing the statement that $P(u)$ has all its rots of size $q^{-1/2}$ (the "Riemann hypothesis") but this is understandable: one needs a fair amount of machinery to prove this. What startled me (in my naivity) was that I had assumed that all this was due to Weil in the 1940s and I am obviously wrong: "all Weil did" was to prove RH. So I have a very basic history question:

Q2) However did Schmidt do this?


EDIT: brief summary of answers below (and what I learned from chasing up the references):

A1) Artin didn't do anything like what I suggested. He could explicitly compute the zeta function of an arbitrary given hyperelliptic curve over a given finite field by an elegant application of quadratic reciprocity. See e.g. the first of Roquette's three papers below. The method in theory works for all genera although the computations quickly get tiresome.

A2) Riemann-Roch. Express the product defining $Z$ as an infinite sum and then use your head.

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This agrees with the Wikipedia article. Weil proved the final step for curves over finite fields. However, F.K. Schmidt is mentioned nowhere in the wikipedia article, and he doesn't have an article himself. –  Harry Gindi Feb 8 '10 at 10:51
    
I haven't done it yet, but in retrospect clearly a sensible way to approach Q2 is to dig out Weil's work on the Weil conjectures. This sounds less daunting than trying to find Artin's work. –  Kevin Buzzard Feb 8 '10 at 10:57
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The rationality and functional equation follow fairly easily from the Riemann-Roch theorem. This is how one usually proves them for elliptic curves, for example. Schmidt's proof can be found in many places,for example, in V Section 5 of Eichler, Introduction to the Theory of Algebraic Numbers and Functions. Artin's work on the zeta function is in his thesis, which can be found most easily in his collected works (which is on my bookshelf, but I don't have time right now). –  JS Milne Feb 8 '10 at 12:12
    
Kevin, note also the curious historical similarity between the development of number fields and function fields: in both settings, the first steps were taken in quadratic extensions before a general theory was built up. –  KConrad Feb 13 '10 at 3:03
    
Yes isn't that great. In fact Artin was motivated by loads of arithmetic questions (class number formulas, quadratic reciprocity etc) and proved analogues of many of them in his thesis in the function field setting. For him QR was important because it implies rationality of the zeta function in the hyperelliptic case! He couldn't use RR because it wasn't there in the finite field setting. I still don't know how he proved the functional equation though (but in some sense I don't care because I can prove it with RR now). –  Kevin Buzzard Feb 13 '10 at 8:21
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2 Answers 2

up vote 15 down vote accepted

Peter Roquette has written three beautiful papers on the history of the zeta-function.

The Riemann hypothesis in characteristic p, its origin and development. Part 1. The formation of the zeta functions of Artin and F.K. Schmidt.

The Riemann hypothesis in characteristic p, its origin and development. Part 2. The first steps by Davenport and Hasse.

The Riemann hypothesis in characteristic p, its origin and development. Part 3: The elliptic case.

Relevant to your questions is part 1. From the abstract " This Part 1 is dealing with the development before Hasse's contributions to the Riemann hypothesis. We are trying to explain what he could build upon. The time interval covered will be roughly between 1921 and 1931. We start with Artin's thesis of 1921 where the Riemann hypothesis for function fields was spelled out and discussed for the first time, namely in the case of quadratic function fields. We will describe the activities following Artin's thesis until F.K.Schmidt's classical paper 1931 on the Riemann-Roch theorem and the zeta function of an arbitrary function field. Finally we will review Hasse's paper in 1934 where he gives a summary about all what was known at that time about zeta functions of function fields. "

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See also MR1240754 (94i:11060) Cartier, Pierre Des nombres premiers à la géométrie algébrique (une brève histoire de la fonction zéta). (French) [From primes to algebraic geometry (a brief history of the zeta function)] Analyse diophantienne et géométrie algébrique, 51--77, Cahiers Sém. Hist. Math. Sér. 2, 3, Univ. Paris VI, Paris, 1993. –  Chandan Singh Dalawat Feb 8 '10 at 13:12
    
I would have accepted Milne's answer because I chased up the reference he gave to that Eichler book and it explained precisely what I was missing. But given that he only made it a comment, I'll vote his comment up but accept your much more thorough answer. Thanks! –  Kevin Buzzard Feb 8 '10 at 19:15
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Roquette's articles contain the whole story; briefly, here are the main facts:

  1. Artin's collected works are quite small, and his thesis (two parts) is put right at the beginning.

  2. Artin looked at quadratic extensions of the rational funtion field and distinguished between real and imaginary extensions; geometrically, these are affine pieces of hyperelliptic curves. The birational point of view, in which these distinctions disappear, was introduced later by F.K. Schmidt. Moreover, Artin talked about ideal classes instead of points on the curves.

  3. The rationality of the zeta functions follows from the quadratic reciprocity law in the rational function field.

  4. Given a specific extension, it is easy to compute the zeta function as well as its zeroes, and verify the Riemann hypothesis. This is what Artin did for extensions of small degree.

  5. F.K. Schmidt introduced "birational invariance" by looking at the projective curves and realized that the rationality can be proved by Riemann-Roch instead of the reciprocity law.

  6. I think Artin started reading Hecke only after his thesis, so he did not know about
    Hecke characters at the time.

Thus Artin verified the Riemann hypothesis for specific examples of quadratic extensions, F.K. Schmidt dervied the functional equation and the rationality for general extensions, Hasse proved the RH for curves of genus 1, and Weil gave the proof for curves (and extended the conjecture to varieties of arbitrary dimension).

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Thanks a lot for your additional comments Franz. In fact you've probably answered my first question with your comment (4), the details of which I've not yet understood. I think I'll look at the Roquette articles before I bug you about it though! –  Kevin Buzzard Feb 8 '10 at 19:19
    
For a curve of genus 1, the zeta function only depends on a_p: you only have to count the number of points modulo p and you know the whole zeta function. Artin discussed curves of degree 3 and 4 over fields ith p = 3, 5 and 7 elements in his thesis an did not verify the RH for genus 2 curves. Originally he had planned to extend his results to arbitrary finite fields, but when Hilbert belittled his efforts during a talk he gave at Goettingen, Artin dropped the subject. He intended to –  Franz Lemmermeyer Feb 8 '10 at 19:40
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